**4. Discussion**

44 New Approaches to the Study of Marine Mammals

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**Figure 9.** Changes in carbohydrate and carbohydrate-derived pool over 24 hours. Apparently the glucose and lactate pools were initialized incorrectly, as reflected in their rapid increase in the first hours of the simulation. Once they became stable, they both declined at the same rate, which was equivalent to the rate of decline of the glycerol pool. This similarity in rates of decline suggests that these pools are closely

Hour 0 Hour 4 Hour 8 Hour 12 Hour 16 Hour 20 Hour 24 **Hours**

**Carbohydrates Through A One-Day Simulation**

**Glucose**

**Glycerol**

**Lactate**

**Figure 10.** Changes in the glucose, glycerol, lactate, alanine and ketones pools over 24 hours after reestimation of the initial conditions by extrapolating the linear parts of Figures 6, 7, and 8 to time zero.

linked through recycling pathways, and may in fact represent a subsystem of the model.

As initially formulated, the value of the rate constant for efflux from the sink pool (k10,10) was set at zero, because there should be no efflux from the sink. Eigenanalysis of this matrix yielded 9 negative eigenvalues and one zero eigenvalue, indicating a stable, closed system (Keith 1999). An eigenvalue of zero is to be expected from a matrix with a zero on the main diagonal (Edelstein-Keshet 1988, Swartzman 1987). For this reason, the zero eigenvalue can be considered an artifact of matrix construction, and not truly representative of the system being simulated. Therefore, a second analysis was conducted in which the value of the rate constant for efflux from the sink pool was set to one, which allowed all of the contents of the sink pool to exit at every time step. Eigenanalysis of this matrix yielded 10 negative eigenvalues, indicative of a now open and still stable system. The trace (τ) of this matrix was -0.186, and the determinant (Δ) was 2.174 x 10-32. In this case τ2 was 0.0346 and 4Δ was 8.696 x 10-32. Taken together, these values indicate that the system without a sink lies near a stable node condition in the phase-plane. However, this is difficult to reconcile with the biological reality that a fasting elephant seal with no food or water inputs is not at equilibrium, and cannot survive forever (Ortiz *et. al.* 1978).

The genesis of this apparent contradiction may lie in differences in time scale or time constants. Differential equations with widely different time scales are "stiff" (Heinrich et al. 1977) and will have eigenvalues of different orders of magnitude. This is apparent here where the eigenvalues range from -1.06 x 10-1 to -2.81 x 10-7. The reciprocals of the eigenvalues are the relaxation times (Heinrich et al. 1977) and these likewise vary over six orders of magnitude, indicating that there are fast-reacting variables and slow-reacting variables in the simulation. Such hierarchical time structure may obscure predictions of model stability because the eigenvalues only characterize the system in the close time-neighborhood of the steady state where linear approximation is appropriate (Heinrich et al. 1977). Thus, predictions of model stability based on the signs of the eigenvalues may contradict a prediction of model instability based on relaxation times and slow-moving versus fast-moving variables in the system (Heinrich et al. 1977) over the duration of the actual fast of the animal.

Elevated palmitate levels are consistent with field data indicating that the major energy substrate during fasting (Castellini, *et. al.* 1987, Keith 1984). The decline in ketone levels through the simulation is consistent with field data indicating low levels of ketone bodies in the plasma of fasting northern elephant seal pups (Costa and Ortiz 1982). Declines in alanine, tissue protein, and urea levels are also consistent with field data (Pernia, *et. al.* 1980), and are validated by data which show that the lean body mass of the animal doesn't change during significantly during the fasting period (Ortiz, *et. al.* 1978). Lack of significant protein catabolism, with concomitant low urea levels, indicates that the animals do not maintain their elevated blood glucose levels at the expense of gluconeogenesis from amino acids (Keith 1984). Close similarities in the rates of decline of the glucose, glycerol, and lactate pools in the later parts of the simulation may be indicative of the extensive glucose carbon recycling which occurs in these animals during fasting. There is extensive interchange of carbon between these three pools, as indicated by high levels of Cori cycle and glucoseglycerol cycle activity (Keith 1984, Keith and Ortiz 1989). It is postulated that this high

degree of recycling may be one reason for the ability of these animals to avoid ketoacidosis, a major deleterious consequence of fasting in many other mammals, and thus allow them to undergo a prolonged fast during this vulnerable period in their life history.

**Section 3** 

**Environmental/Ecological Issues** 
